We divide the proof in several steps.
Step 1 (Compactness) 
Thanks to Lemma REF  and to Proposition REF 
(see in particular ()) we have that the exist
\(\mu \in C^{0,1/2}([0,T];{M}_2(M))\)  such that for any \(t\in [0,T]\)
\(\omega (u_\varepsilon (t)) \rightarrow \mu (t)\qquad \hbox{ in } W^{-1,p}(M) \quad \textrm {for any } p\in (1, \, 2)\) 
Thanks to Proposition REF  for any \(t\in (0,T)\)  the measure \(\mu (t)\)  is of the form
\(\mu (t) = 2\pi \sum _{j=1}^{n(t)}d_j(t) \delta _{a_j(t)}, \,\,\,\hbox{ where } d_j\in \mathbb {Z},\) 
where for any \(t\in [0,T]\)  \(n(t)\in \mathbb {N}\)  and \(a_j(t)\in M\)  for \(j=1,\ldots , n(t)\)  with \(a_j(t)\ne a_k(t)\)  for \(j\ne k\)  and satisfies
\(\liminf _{\varepsilon \rightarrow 0}\frac{F_\varepsilon (u_\varepsilon (t))}{\pi \vert \log \varepsilon \vert }\ge \sum _{j=1}^{n(t)}\vert d_j(t)\vert .\) 
Therefore, since
\(F(u_\varepsilon (t))\le F_\varepsilon (u_\varepsilon ^{0})\le \pi n\vert \log \varepsilon \vert +C,\) 
we conclude that for \(t\in [0,T)\) ,
\(\sum _{j=1}^{n(t)} \vert d_j(t)\vert \le \sum _{j=1}^{n}\vert d_j\vert .\) 
The fact that the evolution decreases the total charge combined with the continuity of \(t\mapsto \int _{M}\mu (t)\zeta \) , for \(\zeta \in C^{1}_c(M)\) ,
give (see [1]}) that there exists \(T^*\in (0,T]\) 
Note that since \(u_\varepsilon \)  can be defined in the whole \((0,+\infty )\) , we can always assume that \(T^*\le T\) .
and \(\mathbf {a}:[0,T^*)\rightarrow M^n\)  with \(\mathbf {a}\in H^1(0,T^*;M^n)\)  such that \(a_j(t)\ne a_k(t)\)  for \(j\ne k\)  and for any \(t\in [0,T^*)\) . Moreover, \(d_j(t) = d_j(0) =\pm 1\)  for \(t\in [0,T^*)\)  and thus
the measure \(\mu \)  has the form
\(\mu (t) =2\pi \sum _{j=1}^n d_j\delta _{a_j(t)}\) 
for any \(t\in [0,T^*)\) .
Therefore, we are in the position to apply Proposition REF  to obtain ().
Finally, we consider the sequence \(\xi _\varepsilon :=\mathbb {P}(j(u_\varepsilon ))\) .
The sequence \(u_\varepsilon \)  verifies all the hypothesis of Proposition REF  and therefore there exists \(\xi \in H^1(0,T^*;\mathrm {Harm}^1(M))\)  such that
\(\mathbb {P}(j(u_\varepsilon (t)))\xrightarrow{} \xi (t)\qquad \hbox{ for any }\quad t\in (0,T^*).\) 

Step 2 (Convergence to a harmonic vector field) 
We have
\(\int _{0}^T\int _{M}\left| \Delta _g u_\varepsilon + \frac{1}{\varepsilon ^2}(\vert 1- \vert u_\varepsilon \vert ^2)u_\varepsilon  \right|^2_{g} \mathrm {vol}_g\mathrm {d}t = \frac{1}{\vert \log \varepsilon \vert ^2}\int _{0}^{T}\int _M\vert \partial _t u_\varepsilon \vert ^2_g \, \mathrm {vol}_g\mathrm {d}t,\) 
and hence, thanks to (REF ),
\(\int _{0}^T \int _{M}\left| \Delta _g u_\varepsilon + \frac{1}{\varepsilon ^2}\left( 1- \left| u_\varepsilon  \right|_g^2\right)u_\varepsilon  \right|_{g}^2 \, \mathrm {vol}_g\mathrm {d}t \le \frac{C}{\vert \log \varepsilon \vert }.\) 
Therefore
\(\Delta _g u_\varepsilon + \frac{1}{\varepsilon ^2}\left(1-\left| u_\varepsilon  \right|_g^2\right)u_\varepsilon \rightarrow 0\,\,\,\,\hbox{ in }L^2(0,T;L^2(M)),\) 
and, up to subsequence,
\(\Delta _g u_\varepsilon + \frac{1}{\varepsilon ^2}\left(1-\left| u_\varepsilon  \right|_g^2\right)u_\varepsilon \rightarrow 0 \qquad \hbox{ in } L^2_{\operatorname{tan}}(M)\quad \hbox{ and for almost any }t\in (0,T^*).\) 
We let \(C\subseteq (0,T^*)\)  the set of those \(t\)  in which the above convergence holds.
We fix a \(t\in C\) .
Thanks to Lemma REF  and to Corollary REF ,
there exists a subsequence \(\varepsilon _h(t)\xrightarrow{} 0\) ,
which may depend on the chosen \(t\) , and a vector field \(u^*(t)\in H^1_{\operatorname{tan},\text{loc}}(M\setminus \left\lbrace a_1(t),\ldots ,a_d(t)\right\rbrace )\cap W^{1,p}_{\operatorname{tan}}(M)\)  such that \(\left| u^*(t) \right|_g=1\)  and
\(& u_{\varepsilon _h(t)}(t) \rightarrow u^*(t)\,\,\,\hbox{ weakly in } H^1_{\operatorname{tan},\text{loc}}(M\setminus \left\lbrace a_1(t),\ldots ,a_d(t)\right\rbrace ),\\& u_{\varepsilon _h(t)}(t)\rightarrow u^*(t)\,\,\,\hbox{ weakly in } W^{1,p}_{\operatorname{tan}}(M)\,\,\,\,p\in [1,2),\\& u_{\varepsilon _h(t)}(t)\rightarrow u^*(t)\,\,\,\hbox{ strongly in } L^p_{\operatorname{tan}}(M)\,\,\,\,p\in [1,+\infty ).\) 

Since \(D u_{\varepsilon _h(t)}(t)\)  converges weakly in \(W^{1,p}_{\operatorname{tan}}(M)\)  for \(p<2\)  and \(u_{\varepsilon _h(t)}(t)\)  converges strongly in \(L^p_{\operatorname{tan}}(M)\)  for any \(p<+\infty \)  we get that
\(j(u_{\varepsilon _h(t)}(t))\rightharpoonup j(u^*(t))\qquad \hbox{ in }\quad L^p(M)\quad \hbox{ for any }p\in [1,2).\) 
Therefore, thanks to (REF ), we have that
\(\mathrm {d}j(u^*(t)) = -\kappa \mathrm {vol}_g+ 2\pi \sum _{j=1}^n d_j\delta _{a_j(t)},\,\,\,\,\,\,\hbox{ for a.a. }t\in [0,T^*).\) 
As a result, we have that
\(0 \stackrel{\text{Stokes}}{=} \int _{M}\mathrm {d}j(u^*(t)) = 2\pi \sum _{j=1}^n d_j -\int _{M}\kappa \mathrm {vol}_g\stackrel{\text{Gauss-Bonnet}}{=} 2\pi \sum _{j=1}^n d_j -2\pi \chi (M),\) 
and thus we have that \((\mathbf {a}(t), \mathbf {d})\in {A}^n\) .
Now we prove that \(\mathrm {d}^* j(u^*(t)) =0\) . This will imply that \(u^*(t)\)  is indeed a canonical harmonic vector field.
To this end we let \(\psi :M\rightarrow \mathbb {R}\)  be a smooth function.
Then, since
\(\frac{1}{\varepsilon ^2}(1-\vert u_{\varepsilon _h(t)}\vert ^2)(u_{\varepsilon _h}(t),i u_{\varepsilon _h(t)})_g\psi = 0\qquad \hbox{ in } M\) 
we readily have that
\(\lim _{h\rightarrow +\infty }\int _{M}\left(\Delta _g u_{\varepsilon _h(t)},iu_{\varepsilon _h(t)}(t)\right)_g\psi \mathrm {vol}_g= 0\) 
and thus, thanks to Lemma REF  below, that
\(\lim _{h\rightarrow +\infty }\int _{M}\psi \,\mathrm {d}^*j(u_{\varepsilon _h(t)}(t)) \mathrm {vol}_g= 0.\) 
Thus, thanks to (REF ), we have
\(0 = \lim _{h\rightarrow +\infty }\int _{M}\psi \,\mathrm {d}^*j(u_{\varepsilon _h(t)}(t)) \mathrm {vol}_g=\lim _{h\rightarrow +\infty }\int _{M}\mathrm {d}\psi \,j(u_{\varepsilon _h(t)}(t)) \mathrm {vol}_g=\int _{M}\mathrm {d}\psi \,j(u^*(t))\mathrm {vol}_g,\) 
namely
\(\mathrm {d}^*j(u^*(t)) = 0.\) 
Thus we conclude that \(u^*(t)\)  is a canonical harmonic vector field.
Since \(t\)  was indeed arbitrary on \(C\)  we have proved that for almost any \(t\in (0,T^*)\)  we have that \((\mathbf {a}(t),\mathbf {d})\in {A}^n\)  with associated a canonical harmonic vector field \(u^*(t)\) .
Note that \(u^*(t)\)  is obtained as a limit of \(u_\varepsilon (t)\)  along a selected subsequence.
The continuity of \(\mathbf {a}: [0,T^*)\rightarrow M^n\)  actually implies that \((\mathbf {a}(t), \mathbf {d})\in {A}\)  for any \(t\in [0,T^*)\) .
The fact that at time \(t\in C\)  the vector field \(u^*(t)\)  is a canonical harmonic vector field implies that the harmonic 1-form \(\xi (t)\in \mathcal {L}(\mathbf {a}(t), \mathbf {d})\) . Indeed, on the one hand the convergences (REF ) and (REF ) imply that
\(\xi (t) = \mathbb {P}j(u^*(t)).\) 
On the other hand
we let \(\Phi _t=\Phi (\mathbf {a}(t),\mathbf {d})\)  be the 2-form such that
\({\left\lbrace \begin{array}{ll}-\Delta \Phi = -\kappa \mathrm {vol}_g+ 2\pi \displaystyle \sum _{j=1}^n d_j\delta _{a_j(t)}\\\displaystyle \int _{M}\Phi =0.\end{array}\right.}\) 
Then,
(REF ) and (REF )
imply that the 1-form \(j(u^*(t))- d^*\Phi _t \in \text{Harm}^1(M)\) . Therefore [2]}, and the uniqueness of the projection imply that
\(\xi (t) = j(u^*(t))- \mathrm {d}^*\Phi _t, \qquad \xi (t) \in \mathcal {L}(\mathbf {a}(t), \mathbf {d})\quad \text{ for almost any }t\in [0,T^*).\) 
The continuity of \(\xi :[0,T^*)\rightarrow \mathrm {Harm}^1(M)\)  and of \(\mathbf {a}:[0,T^*)\rightarrow M^n\)  and the continuity of the lattice \(\mathcal {L}\)  with respect to \((\mathbf {a},\mathbf {d})\)  (see [2]}) give that
\(\xi (t)\in \mathcal {L}(\mathbf {a}(t),\mathbf {d})\qquad \hbox{ for any }\quad t\in [0,T^*).\) 
Therefore, [2]} gives that
for any \(t\in [0,T]\)  there exists a canonical harmonic vector field \(u^*(t)\) .

